Two corners of a triangle have angles of #pi / 3 # and # pi / 6 #. If one side of the triangle has a length of #4 #, what is the longest possible perimeter of the triangle?

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Answer 1 · 2016-12-01T10:57:11

The maximum perimeter is #P=12+4sqrt(3)#

As the sum of the internal angles of a triangle is always #pi#, if two angles are #pi/3# and #pi/6# the third angle equals:

#pi-pi/6-pi/3 = pi/2#

So this is a right triangle and if #H# is the length of the hypotenuse,
the two legs are:

#A=Hsin(pi/6)=H/2#
#B = Hsin(pi/3)=Hsqrt(3)/2#

The perimeter is maximum if the side length we have is the shortest of the three, and as evidenty #A < B < H# then:

#A=4#
#H=8#
#B=4sqrt(3)#

And the maximum perimeter is:

#P=A+B+H=12+4sqrt(3)#